The CRIX is a market index and follows for the derivation the Laspeyres Index. The index of Laspeyres is defined
as

with Pit the price of the crypto i at time point t and Qi0 the amount of crypto I at time point 0. Pi0 is the price at time point 0. The final index formula is a modification
of the Laspeyres index, given by

where MVit is the market capitalization of the crypto i at time point t. The Divisor ensures that the changes in the
amount of coins of a crypto doesn’t affect the value of the CRIX. At the starting point of the CRIX is the Divisor chosen such that

The starting value of
the CRIX is therefore chosen to be 1000. Whenever the amount of coins of a
crypto changes, the Divisor has to be adjusted. This shall ensure that just price changes cause changes in the
value of the CRIX. The applied formula is

Divisort-1 is the Divisor directly in front of the change in
the amount of coins and Divisort directly afterwards. Therefore, it
is ensured that the CRIX starts again with the same value than directly before the change in the amount of
coins.

Liquidity rule: It can happen, that a crypto has a high market capitalization and therefore
shall be part of the CRIX. But if it is not traded frequently, it can’t be easily converted to traditional fiat
money or other cryptocurrencies. On top of this drawback, a low relative trading volume to the rest of the asset
universe shows that the users care more for other cryptos. In a representative benchmark shall be just included
the cryptos which are used often. Two measures will be applied which are modified versions of the liquidity
rules from the STOXX Japan 600 and the AEX Family. From the first is taken the approach to look at the
Percentiles of the Average Daily Trading Volume (ADTV) and from the later the concept to take
also the number of shares into account. The applied rules are the following:

0.25 percentile of ADTV

where ADTV0.25 is the 0.25 percentile of the ADTVs of all cryptos in the last period and ADTVi is the ADTV of a single crypto. The approach
ensures that just cryptos which are liquid in relation to the other cryptos are eligible for the CRIX. The 0.25 percentile is chosen as rule of thumb
border for the tails of the empirical distribution because this value serves as an important lower bound in
statistical analysis, e.g. the Box-plot. It ensures that the relatively less traded cryptos are excluded
from the eligible ones for CRIX.

0.25 percentile of Average Daily Traded Coins (ADTC)

where ADTC0.25 is the 0.25 percentile of the ADTCs of all cryptos in the last period and ADTCi is the
ADTC of a single crypto. Since many coins
have a very small price, is it likely that they will be excluded by the first rule from the eligible ones even
when many coins are traded but volume will be still low due to the low price. But on the other hand, it doesn’t
pay out to look just at the ADTC since cryptos with a high price will be
possibly less traded but attract much liquidity, e.g. Bitcoin.

Finally, if a crypto fulfills at least one of the two rules, it is eligible for the CRIX.

Number of Constituents:

To our knowledge are the number of constituents of an index normally fixed. This may be a good approach for
relatively stable markets but the crypto market is a very fast and innovative one. To be able to find always the
best benchmark, is it better to variate the amount of member such that always the best benchmark is provided.
The used procedure for CRIX is based on the AIC and BIC criterion, which are the Akaike Information Criterion
and the Schwartz Information Criterion respectively. First will be computed an index representing the total
market by applying the CRIX formula with the necessary capping. But the Liquidity
rule won’t be used so that really all cryptos are included. Afterwards will several indices with
different numbers of constituents be computed. The index representing the total market will be treated as the
observed values for the market situation and be compared against the indices to check which one provides the
best approximation while using as less index members as possible. The Likelihood function for computing the AIC
and BIC is based on stable distributions to ensure that a broad band of distributions can be covered. For more
details, see the presentation in the references.

Weights:

Each cryptos in CRIX is weighted with its market capitalization.

Ranking:

After excluding cryptos which doesn’t fulfill the Liquidity rule will the cryptos be ranked by
their market capitalization in the last period from highest to lowest. The number of cryptos, obtained from the
rule for the Number of Constituents, with the highest market capitalization will be taken.

Reallocation:

The reallocation period of the CRIX is 1 month to ensure that the index is always up to date. This period is
taken rather short such that the CRIX can react faster to changes in the crypto market. At this time point will
be checked again the liquidity of the cryptos by taking into account the last month. Furthermore will be builded
again the Ranking and afterwards are computed the weights of the cryptos by taking into account
the rule described in Weights. On top of that will the Divisor be adjusted.

Every 3 month will be checked if the number of constituents still fit the market well. For the computation will
be used the values from the last 3 month to ensure that just the most recent values influence this important
decision for the CRIX.

Special Events

If the current price for a crypto which is part of CRIX is not reachable from the exchanges is the CRIX
made insensitive to its value as long as the value can’t be reached.

If a crypto vanished from the market while it is part of CRIX while its value be made insensitive until
the next reallocation date is reached.

VCRIX:

VCRIX (based on CRIX) is a volatility index, able to grasp the risk induced by the crypto-currency market, much like VIX for the S&P components. The crypto ‘fear index’ addresses the limitations constituted by the absence of developed derivative market for cryptos and offers an adequate measure for implied volatility, thus proved to be a proper basis for option pricing.